---
tag: X60JG9AEQJ
kind: concept
name: Kähler manifold
msc: [32Q15, 53C55]
specializes: [CK1DCPC3D3]
sources:
  - url: https://en.wikipedia.org/wiki/K%C3%A4hler_manifold
same_as:
  wikidata: Q1353916
  wikipedia: Kähler manifold
---

## Summary

A Kähler manifold is a [[complex manifold|JSGWAN2RJV]] with a Hermitian metric whose associated $2$-form is closed.
Its Riemannian, complex and symplectic structures are then compatible in the strongest way.
Complex projective space and all its complex submanifolds are Kähler, so Kähler geometry contains smooth projective algebraic geometry.
Compact Kähler manifolds satisfy strong topological restrictions, such as the Hodge decomposition of their cohomology and the evenness of odd Betti numbers.

## Explanations

### L1 human

On a complex manifold one can measure lengths with a Hermitian metric, and every such metric has a $2$-form attached to it.
A Kähler metric is one where this $2$-form has zero derivative.
This condition means that, to second order at each point, the metric looks like the flat metric of $\C^n$ in suitable holomorphic coordinates.
The flat space $\C^n$, every Riemann surface and complex projective space with its Fubini–Study metric are Kähler.
Many compact complex manifolds, such as Hopf manifolds, are not.